132 The interaction of atoms with radiation
energy−μ·Bis constant, and the magnetic moment precesses around
the direction of the fieldB=B̂ez. In the Bloch description the fictitious
magnetic field lies alongWand the magnitudeWin eqn 7.28 determines
the precession rate.Example 7.1 Resonant excitation (δ =0)givesW=Ω̂e 1 andR
describes a cone about̂e 1. An important case is when all the population
starts in level 1 so that initiallyR·̂e 1 = 0; in this case the Bloch vector
rotates in the plane perpendicular tôe 1 mapping out a great circle on
the Bloch sphere, as drawn in Fig. 7.2(c). This motion corresponds to
the Rabi oscillations (eqn 7.29). In this picture aπ/2-pulse rotates the
Bloch vector throughπ/2about̂e 1. A sequence of twoπ/2-pulses gives
aπ-pulse that rotates the Bloch vector (clockwise) throughπabout
̂e 1 ,e.g.w=1→w=−1 and this represents the transfer of all the(^17) In this particular example the fi- population from level 1 to 2. (^17) This is consistent with the more general
nal state is obvious by inspection, but
clearly the same principles apply to
other initial states, e.g. states of the
form
{
| 1 〉+eiφ| 2 〉
}
/
√
2 that lie on the
equator of the sphere. The Bloch
sphere is indispensable for thinking
about more complex pulse sequences,
such as those used in nuclear magnetic
resonance (NMR).
statement given in eqn 7.30.
The very brief introduction to the Bloch sphere given in this section
shows clearly that a two-level atom’s response to radiation does not
increase indefinitely with the driving field—beyond a certain point an
increase in the applied field (or the interaction time) does not produce
a larger dipole moment or change in population. This ‘saturation’ has
important consequences and makes the two-level system different from a
classical oscillator (where the dipole moment is proportional to the field,
as will be shown in Section 7.5).
7.4 Ramsey fringes
The previous sections in this chapter have shown how to calculate the
response of a two-level atom to radiation. In this section we shall apply
this theory to radio-frequency spectroscopy, e.g. the method of magnetic
resonance in an atomic beam described in Chapter 6. However, the same
principles are important whenever line width is limited by the finite
interaction time, both within atomic physics and more generally. In
particular, we shall calculate what happens to an atom subjected to two
pulses of radiation since such a double-pulse sequence has favourable
properties for precision measurements.
An atom that interacts with a square pulse of radiation, i.e. an os-
cillating electric field of constant amplitude from timet=0toτp,and(^18) This assumes weak excitation: E 0 = 0 otherwise, has a probability of excitation as in eqn 7.15. (^18) This
|c 2 |^2 1. excitation probability is plotted in Fig. 7.1 as a function of the radia-
tion’s frequency detuning from the (angular) resonance frequencyω 0 .As
stated below eqn 7.16, the frequency spread given by the first minimum
(^19) This is not the FWHM but it is close of the sinc (^2) function corresponds to a width 19
enough for our purposes.
∆f=
∆ω
2 π
=
1
τp. (7.50)
The frequency spread is inversely proportional to the interaction time,^20(^20) This expression is equivalent to
eqn 6.40 that was used to calculate the
line width for an atomic clock.